Hostname: page-component-cd9895bd7-gxg78 Total loading time: 0 Render date: 2024-12-23T14:15:35.809Z Has data issue: false hasContentIssue false
  • English
  • Français

Extremal and additive processes generated by Pareto distributed random vectors

Published online by Cambridge University Press:  15 October 2014

Kosto V. Mitov  and
Saralees Nadarajah
Kosto V. Mitov
Affiliation:
Aviation Faculty, NMU, 5856 D. Mitropolia, Pleven, Bulgaria. [email protected]
Saralees Nadarajah
Affiliation:
School of Mathematics, University of Manchester, Manchester, M13 9PL, UK; [email protected]
Get access

Abstract

Pareto distributions are most popular for modeling heavy tailed data. Here, we obtain weak limits of a sequence of extremal and a sequence of additive processes constructed by a series of Bernoulli point processes with bivariate Pareto space components. For the limiting processes we derive the one dimensional distributions in explicit forms. Some of the main properties of these distributions are also proved.

Keywords

Additive processextremal processlimit theoremspareto distribution
Type
Research Article
Information
ESAIM: Probability and Statistics , Volume 18 , 2014 , pp. 667 - 685
Copyright
© EDP Sciences, SMAI, 2014

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Arnold, B.C. and Press, S.J., Bayesian-inference for Pareto populations. J. Econom. 21 (1983) 287–306. Google Scholar
Balkema, A. and Pancheva, E., Decomposition of multivariate extremal processes. Commun. Stat Theory Methods 25 (1996) 737–758. Google Scholar
Barnett, V., Some outlier tests for multivariate samples. South African Stat. J. 13 (1979) 29–52. Google Scholar
N. Bingham, C. Goldie and J. Teugels, Regular Variation. Cambridge University Press, Cambridge (1987).
Bouyé, E., Multivariate extremes at work for portfolio risk measurement. Finance 23 (2002) 125–144. Google Scholar
Cebrián, A.C., Denuit, M. and Lambert, P., Analysis of bivariate tail dependence using extreme value copulas: An application to the SOA medical large claims database. Belgian Actuarial Bull. 3 (2003) 33–41. Google Scholar
Coles, S.G. and Tawn, J.A., Statistical methods for multivariate extremes: An application to structural design (with discussion). J. Appl. Stat. 43 (1994) 1–48. Google Scholar
Demarta, S. and McNeil, A.J., The t copula and related copulas. Int. Stat. Rev. 73 (2005) 111–129. Google Scholar
A. Dias and P. Embrechts, Dynamic copula models for multivariate high-frequency data in finance. Working Paper, ETH Zurich (2003).
P. Embrechts and M. Maejima, Self-similar Processes. Princeton University Press, Princeton (2002).
Eryilmaz, S. and Iscioglu, F., Reliability evaluation for a multi-state system under stress-strength setup. Commun. Stat. Theory Methods 40 (2011) 547–558. Google Scholar
Fawcett, L. and Walshaw, D., Markov chain models for extreme wind speeds. Environmetrics 17 (2006) 795–809. Google Scholar
Galambos, J., Order statistics of samples from multivariate distributions. J. Amer. Stat. Assoc. 70 (1975) 674–680. Google Scholar
Ghorbel, A. and Trabelsi, A., Measure of financial risk using conditional extreme value copulas with EVT margins. J. Risk 11 (2009) 51–85. Google Scholar
I.S. Gradshteyn and I.M. Ryzhik, Table of Integrals, Series, and Products, 7th edition. Academic Press, San Diego (2007).
Hürlimann, W., Fitting bivariate cumulative returns with copulas. Comput. Stat. Data Anal. 45 (2004) 355–372. Google Scholar
Hüsler, J. and Reiss, R.-D., Maxima of normal random vectors: Between independence and complete dependence. Stat. Probab. Lett. 7 (1989) 283–286. Google Scholar
Hutchinson, T.P., Latent structure models applied to the joint distribution of drivers’ injuries in road accidents. Stat. Neerlandica 31 (1977) 105–111. Google Scholar
Hutchinson, T.P. and Satterthwaite, S.P. (1977). Mathematical-models for describing clustering of sociopathy and hysteria in families. British J. Psychiatry 130 294–297. Google ScholarPubMed
Jansen, D., and de Vries, C., On the frequency of large stock market returns: Putting booms and busts into perspective. Rev. Econ. Stat. 23 (1991) 18–24. Google Scholar
S. Jäschke, Estimation of risk measures in energy portfolios using modern copula techniques. Discussion Paper No. 43, Dortmund (2012).
Joe, H., and Li, H., Tail risk of multivariate regular variation. Methodology Comput. Appl. Probab. 13 (2011) 671–693. Google Scholar
Joe, H., Smith, R.L. and Weissman, I., Bivariate threshold methods for extremes. J. R. Stat. Soc. B 54 (1992) 171–183. Google Scholar
R.B. Langrin, Measuring extreme cross-market dependence for risk management: The case of Jamaican equity and foreign exchange markets. Financial Stability Department, Research and Economic Program. Division, Bank of Jamaica (2004).
L. Lescourret and C. Robert, Estimating the probability of two dependent catastrophic events. ASTIN Colloquium. International Acturial Association, Brussels, Belgium (2004).
Lescourret, L., and Robert, C.Y., Extreme dependence of multivariate catastrophic losses. Scandinavian Actuarial J. (2006) 203–225. Google Scholar
Lim, K.-G., Global financial risks, CVaR and contagion management. J. Business Policy Res. 7 (2012) 115–130. Google Scholar
Lindley, D.V. and Singpurwalla, N.D., Multivariate distributions for the life lengths of components of a system sharing a common environment. J. Appl. Probab. 23 (1986) 418–431. Google Scholar
Luo, X. and Shevchenkoa, P.V., The t copula with multiple parameters of degrees of freedom: Bivariate characteristics and application to risk management. Quant. Finance 10 (2010) 1039–1054. Google Scholar
Ma, M., Song, S., Ren, L., Jiang, S. and Song, J., Multivariate drought characteristics using trivariate Gaussian and Student t copulas. Hydrological Proc. 27 (2013) 1175–1190. Google Scholar
Mardia, K.V., Multivariate Pareto distributions. Ann. Math. Stat. 33 (1962) 1008–1015. Google Scholar
Mcgrath, M.F., Gross, D. and Singpurwalla, N.D., A subjective Bayesian approach to the theory of queues I Modeling. Queueing Systems 1 (1987) 317–333. Google Scholar
Meerschaert, M.M., and Scalas, E., Coupled continuous time random walks in finance. Phys. A: Stat. Mech. Appl. 370 (2006) 114–118. Google Scholar
M.M. Meerschaert and H.-P. Scheffler, Limit Distributions for Sums of Independent Random Vectors: Heavy Tails Theory Practice. Wiley, New York (2001).
M.M. Meerschaert and H.-P. Scheffler, Portfolio modeling with heavy tailed random vectors, in Handbook of Heavy-Tailed Distributions in Finance, edited by S.T. Rachev. Elsevier, New York (2003) 595–640.
Mendes, B.V.M. and Moretti, A.R., Improving financial risk assessment through dependency. Stat. Model. 2 (2002) 103–122. Google Scholar
I. Mitov, S. Rachev and F. Fabozzi, Approximation of aggregate and extremal losses within the very heavy tails framework. Technical Report, University of Karlsrhue, University of California, Santa Barbara, submitted to Quant. Finance (2008).
Mohsin, M., Spöck, G. and Pilz, J., On the performance of a new bivariate pseudo Pareto distribution with application to drought data. Stochastic Environmental Research and Risk Assessment 26 (2011) 925–945. Google Scholar
Motamedi, M. and Liang, R.Y., Probabilistic landslide hazard assessment using Copula modeling technique. Landslides 11 (2013) 565–573. Google Scholar
Nazemi, A. and Elshorbagy, A., Application of copula modelling to the performance assessment of reconstructed watersheds. Stochastic Environmental Research and Risk Assessment 26 (2013) 189–205. Google Scholar
Ng, M.W. and Lo, H.K., Regional air quality conformity in transportation networks with stochastic dependencies: A theoretical copula-based model. Networks and Spatial Economics 13 (2013) 373–397. Google Scholar
Pancheva, E. and Jordanova, P., A functional extremal criterion. J. Math. Sci. 121 (2004) 2636–2644. Google Scholar
Pancheva, E. and Jordanova, P., Functional transfer theorems for maxima of iid random variables. C. R. Acad. Bulgare Sci. 57 (2004b) 9–14. Google Scholar
Pancheva, E., Kolkovska, E. and Jordanova, P., Random time-changed extremal processes. Theory Probab. Appl. 51 (2006) 752–772. Google Scholar
Pancheva, E., Mitov, I. and Volkovich, Z., Sum and extremal processes over explosion area. C. R. Acad. Bulgare Sci. 59 (2006) 19–26. Google Scholar
Pancheva, E., Mitov, I. and Volkovich, Z., Relationship between extremal and sum processes generated by the same point process. Serdika 35 (2009) 169–194. Google Scholar
Papadakis, E.N. and Tsionas, E.G., Multivariate Pareto distributions: Inference and financial applications. Commun. Stat. Theory Methods 39 (2010) 1013–1025. Google Scholar
Pickands, J., Multivariate extreme value distributions (with a discussion). In: Proc. of the 43rd Session of the International Statistical Institute, Bull. Int. Stat. Institute 49 (1981) 859–878, 894–902. Google Scholar
A.P. Prudnikov, Y.A. Brychkov and O.I. Marichev, Integrals and Series, vols. 1, 2 and 3. Gordon and Breach Science Publishers, Amsterdam (1986).
S. Rachev and S. Mittnik, Stable Paretian Models in Finance. Wiley, Chichester (2000).
Rosco, J.F. and Joe, H., Measures of tail asymmetry for bivariate copulas. Stat. Papers 54 (2013) 709–726. Google Scholar
Tawn, J.A., Bivariate extreme value theory: Models and estimation. Biometrika 75 (1988) 397–415. Google Scholar
T. Tokarczyk and W. Jakubowsk, Temporal and spatial variability of drought in mountain catchments of the Nysa Klodzka basin. In: Climate Variability and Change – Hydrological Impacts, vol 308, Proc. of 15th FRIEND world conference held at Havana. Edited by S. Demuth, A. Gustard, E. Planos, F. Scatena and E. Servat. 308 (2006) 139–144.
Yang, X., Frees, E.W. and Zhang, Z., A generalized beta copula with applications in modeling multivariate long-tailed data. Insurance: Math. Econ. 49 (2011) 265–284. Google Scholar
Youngren, M.A., Dependence in target element detections induced by the environment. Naval Research Logistics 38 (1991) 567–577. Google Scholar
Yue, S., The Gumbel mixed model applied to storm frequency analysis. Water Resources Management 14 (2000) 377–389. Google Scholar
Zhang, Q., Singh, V.P., Lia, J., Jiang, F. and Bai, Y., Spatio-temporal variations of precipitation extremes in Xinjiang, China. J. Hydrology 434-435 (2012) 7–18. Google Scholar